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DavidMay 27 '14 at 15:26

If you leave copies of your emails on your email account provider's server, check to see if your mail box is full, i.e., you have used up your allowance.
If so delete some that you don't need (or archive them if you can) and this may solve your problem.

If your mail provider is gmail, then it's downloading emails from the "All Mail" folder. It has a copy of every email in your gmail account. When you send or receive an email, the email is received by gmail, it then copies it into the All Mail folder.

It is my understanding that if you use archive instead of delete, it also puts the emails in here instead of the trash.

Are there any messages in your spam/junk mail folder? These might be triggering the incoming mail message but then as they're moved straight away you don't see them.

Do you have any filters (or whatever they're called in Apple mail) active that move messages to other folders? This is less likely given that you state you've checked other folders. (I've seen this behaviour in Outlook for that reason.)